For an ellipse with eccentricity $e = \frac{1}{2}$,the centre is at the origin. If one directrix is $x = 4$,then the equation of the ellipse is

Match the thermodynamic processes taking place in a system with the correct conditions. In the table: $\Delta Q$ is the heat supplied,$\Delta W$ is the work done,and $\Delta U$ is the change in internal energy of the system.

Process	Condition
$(I)$ Adiabatic	$(A) \Delta W = 0$
$(II)$ Isothermal	$(B) \Delta Q = 0$
$(III)$ Isochoric	$(C) \Delta U \neq 0, \Delta W \neq 0, \Delta Q \neq 0$
$(IV)$ Isobaric	$(D) \Delta U = 0$

An insulated container contains $4 \, \text{moles}$ of an ideal diatomic gas at temperature $T$. Heat $Q$ is supplied to this gas, due to which $2 \, \text{moles}$ of gas are dissociated into atoms, but the temperature of the gas remains constant. Then:

Let $f$ be a differentiable function satisfying $f(x + 2y) = 2yf(x) + xf(y) - 3xy + 1$ for all $x, y \in R$ such that $f'(0) = 1$. Then $f(2)$ is equal to:

Read the following five statements and answer the question.
$(a)$ In $Equisetum$ the female gametophyte is retained on the parent sporophyte.
$(b)$ In $Ginkgo$ male gametophyte is not independent.
$(c)$ The sporophyte in $Riccia$ is more developed than that in $Polytrichum$.
$(d)$ Sexual reproduction in $Volvox$ is isogamous.
$(e)$ The spores of slime molds lack cell walls.
How many of the above statements are correct?

$A$ $TV$ transmitting antenna is $128 \ m$ tall. If the receiving antenna is at the ground level,the maximum distance between them for satisfactory communication in line-of-sight mode is (radius of the earth $R_e = 6.4 \times 10^6 \ m$):